Inverse proportion
When one quantity is multiplied by n, the other is divided by n.
How long on a bicycle?
From Lucknow to Kanpur is 90 km. Walking at 5 km/h takes 18 hours. A bicycle goes at 15 km/h, which is 3 times as fast as walking.
How many hours will the trip take on a bicycle?
What this lesson covers
The idea
Two quantities are inversely proportional when they change by the same factor but in opposite directions: when one changes by a factor n, the other changes by the inverse 1/n.
How long on a bicycle?
From Lucknow to Kanpur is 90 km. Walking at 5 km/h takes 18 hours. A bicycle goes at 15 km/h, which is 3 times as fast as walking.
How many hours will the trip take on a bicycle?
- 6 hours
- 15 hours
- 54 hours
Switch the way of travelling
Each hour, the traveller makes one hop as long as the speed. Switch between the ways of travelling. The old trip stays above as a grey row. Look at how the speed changes and how the time changes.
The same factor, the opposite way
Two quantities are inversely proportional when they change by the same factor but in opposite directions: when one changes by a factor n, the other changes by the inverse 1/n.
Walk → bicycle: speed 5 → 15, that is × 3 time 18 → 6, that is × 1/3 The speed went up 3 times, the time came down 3 times.
The same happens with workers. 20 workers take 4 days to lay a road. With half the workers, the work takes double the days: workers 20 → 10, that is × ½ days 4 → 8, that is × 2
In a direct proportion both quantities go up together. In an inverse proportion, when one goes up the other goes down.
Notes
Two quantities are inversely proportional when they change by the same factor but in opposite directions: when one changes by a factor n, the other changes by the inverse 1/n.
Check yourself
The motorcycle goes at 30 km/h and takes 3 hours. A car goes 2 times as fast. How many hours does the car take?
Answer: 1.5 hours
The speed is multiplied by 2, so the time is multiplied by 1/2: 3 ÷ 2 = 1.5 hours. (Check: 60 km/h × 1.5 h = 90 km.)
20 workers take 4 days to lay a road. How many days will 5 workers take to lay the same road?
Answer: 16 days
20 → 5 workers is ÷ 4, so the days are × 4: 4 × 4 = 16 days.
Which two quantities are inversely proportional?
A car at 60 km/h takes 1.5 hours to cover 90 km. How many hours will it take at 20 km/h?
Answer: 4.5 hours
The speed is divided by 3, so the time is multiplied by 3: 1.5 × 3 = 4.5 hours. (Check: 20 km/h × 4.5 h = 90 km.)
- The kilograms of rice and the money paid for it. More rice costs more money: both go up together. That is a direct proportion.
- The hours worked and the money earned at ₹100 an hour. More hours earn more money: both go up together. That is a direct proportion.
- The number of taps filling the same tank and the time to fill it — correct. Yes! 2 times as many taps fill the tank in half the time. One goes up, the other comes down by the same factor.
- The side of a square and its perimeter. When the side doubles, the perimeter also doubles. That is a direct proportion.